MATH 565X Supplementary Homework
Spring 2013
12. Consider the underdetermined linear system Ax = b where A is an m × n matrix with
m < n and b ∈ Rm . Assume the rows of A are linearly independent, so the rank of A equals
the row rank of A equals m. There will then exist infinitely many solutions of the system,
forming an n − m dimensional hyperplane in Rn . The minimum norm solution of Ax = b is
the solution closest to the origin, which may be regarded as the solution of the constrained
optimization problem
min ||x||2
Ax=b
a) Using the Lagrange multiplier method, derive the solution formula x = AT (AAT )−1 b.
b) Find the minimum norm solution of the 3 × 5 system Ax = b when


 
1
2 0 3 2
4



A = −1 −1 4 2 0
b= 1 
3 −2 2 1 1
−7
c) In Matlab, does x = A\b provide the minimum norm solution of Ax = b? If not, find
out how to get it.